A clean chain is a happy chain. It is a good practice to clean your chain periodically when it is on the trainer for the winter. Though there is no mud and water in your basement there are still enough air born particles to gunk up your chain. As a chain wears, contaminants (either air born or from the metal breakdown) will mix with the lube and create an abrasive mixture, leading to premature wear. Cleaning the chain will remove these contaminants and provide a quiet, efficient chain . A chain that is well taken care will last significantly longer and prevent wear to the entire drivetrain. It is always a good idea to check the life of the chain before putting many miles on the trainer unless you plan to change the whole drivetrain at once.